Managing High-Tech Capacity via Managing High-Tech Capacity via Reservation Contracts Reservation Contracts

S. David WuS. David Wu

Lee A. Iacocca Professor and ChairmanLee A. Iacocca Professor and Chairman

Department of Industrial and Systems EngineeringDepartment of Industrial and Systems Engineering

Lehigh UniversityLehigh University

Pennsylvania, U. S. A.Pennsylvania, U. S. A.

*Joint work with Murat Erkoc

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An U.S.Telecommunications An U.S.Telecommunications Component-ManufacturerComponent-Manufacturer

Client Systems

Enterprise/Metro/ Wireless Access

Metro/Regional Transport

Long-Haul Backbone

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High-Tech Manufacturing Supply ChainHigh-Tech Manufacturing Supply Chain

Rapid innovation & volatile demandRapid innovation & volatile demand

High obsolescence rate on technologyHigh obsolescence rate on technology

Short product life-cycle (avg. 18 months)Short product life-cycle (avg. 18 months)

Capital intensive facilities ($2 Billion +)Capital intensive facilities ($2 Billion +)

High capacity costsHigh capacity costs

Manufacturers are conservative in capacity Manufacturers are conservative in capacity expansion of any formexpansion of any form

Capacity reservation contractsCapacity reservation contracts

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Some BackgroundSome Background

Capacity Planning vs. Capacity ReservationCapacity Planning vs. Capacity ReservationOptimization vs. CharacterizationOptimization vs. CharacterizationSolution vs. ProcessSolution vs. Process

The main motivations? The main motivations? how to price capacity reservation? how to price capacity reservation? to go over the thought process analyticallyto go over the thought process analytically preparing for the “soft capacity” environmentpreparing for the “soft capacity” environment

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CM 1Buyer1

BuyerM

Buyer2

Spot Market

CM 2

CM n Supplier K

Supplier 2

Contract Manufacturers

High-Tech Supply Chain Building BlocksHigh-Tech Supply Chain Building Blocks

Capacity Capacity Reservation Reservation

ContractContract

Supplier 1

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Capacity ReservationCapacity Reservation::Settings and Assumptions Settings and Assumptions (Erkoc and Wu, 2002a)(Erkoc and Wu, 2002a)

Between the supplier and a major customerBetween the supplier and a major customer

Stochastic market demandStochastic market demand

Capacity must be built in advance (tactical)Capacity must be built in advance (tactical) outsourcing, work force expansion, procurement of raw outsourcing, work force expansion, procurement of raw

material, or pre-processing of semi-finished productsmaterial, or pre-processing of semi-finished products

Convex capacity cost (considers residual value)Convex capacity cost (considers residual value)

No inventory carry-over (single period)No inventory carry-over (single period)

Complete information (cost, demand distribution)Complete information (cost, demand distribution)

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Product Life-CycleProduct Life-Cycle

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Related Literature Related Literature

Buyer’s perspective: Silver and Jan (1994), Jan and Silver Buyer’s perspective: Silver and Jan (1994), Jan and Silver (1995), Brown and Lee (1998)(1995), Brown and Lee (1998)Supplier-Buyer perspectives: Cachon and Larivier (2001), Supplier-Buyer perspectives: Cachon and Larivier (2001), Tomlin (1999), Serel et. al. (2001)Tomlin (1999), Serel et. al. (2001)Options: Donohue (2000), Barnes-Schuster et. al. (2000) Options: Donohue (2000), Barnes-Schuster et. al. (2000) Main DistinctionsMain Distinctions to reserve or not to reserve to reserve or not to reserve wholesale price is not a contract parameterwholesale price is not a contract parameter linear vs. convex capacity costlinear vs. convex capacity cost forced vs. voluntary complianceforced vs. voluntary compliance buyer vs. supplier led channelbuyer vs. supplier led channel

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Integrated ChannelIntegrated Channel

Supplier

w p

c V(k)

Buyer Market

F(x)

k

F dxxFkkS0

)()(expected sales given capacity k

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Delegation of Control in the Supply ChainDelegation of Control in the Supply Chain

System’s Profit (concave)System’s Profit (concave)

Let Let kkoo be the system optimal solutionbe the system optimal solution

Supplier’s newsvendor profit functionSupplier’s newsvendor profit function

Let Let kk** be the newsvendor solutionbe the newsvendor solution

)()(*)( kVkScp FI

)()(*)(0 kVkScw Fs

)(*

1*

cwFk

)(1

cpFk

oo

okk *

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Double MarginalizationDouble Marginalization- - suboptimalsuboptimal

Supplier

w p

c V(k)

Buyer Market

F(x)

Due to Due to double marginalizationdouble marginalization the system optimal capacity is the system optimal capacity is always greater than the supplier’s always greater than the supplier’s newsvendor capacitynewsvendor capacity

Optimal

Competitive Outcome

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Reservation Contract with Deductible FeeReservation Contract with Deductible Fee

For each unit of capacity reserved, the buyer is charged a feeFor each unit of capacity reserved, the buyer is charged a fee

The fee is later deducted from the purchasing price if the reserved The fee is later deducted from the purchasing price if the reserved capacity is used by the intended customercapacity is used by the intended customer

If the buyer’s demand exceeds the reservation amount, the orders If the buyer’s demand exceeds the reservation amount, the orders are satisfied based on the availability of “extra capacity”are satisfied based on the availability of “extra capacity”

Customer reservations are locked in and some extra capacity may Customer reservations are locked in and some extra capacity may be built during expansionbe built during expansion

The wholesale price The wholesale price w w and product cost and product cost cc are set at the “design- are set at the “design-win” phase, which are not negotiablewin” phase, which are not negotiable

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The Sequence of EventsThe Sequence of Events

Supplier announces reservation fee (r) Buyer announces how

much to reserve (q)

Supplier decides how much capacity to build (k)

Demand is realized (x)

Buyer is penalized for unused reserved capacity

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Supplier’s Capacity DecisionSupplier’s Capacity Decision

If , then the constraint must be bindingIf , then the constraint must be binding

qk

xqErkVkScw Fk

s.t.

),0max()()()( Max

*kq

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Buyer’s ProfitBuyer’s Profit

Buyer profit function is conditional due to the fact that he Buyer profit function is conditional due to the fact that he never reserves less than never reserves less than k*k*

wo

kqB

/)S(kw)(p

ifdxF(x)rS(q)w)(p

*

*q

0

0

• In optimality

• Reservation quantity q decreases in reservation feerwp

wpqF

)( *

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Buyer’s Incentive to ReserveBuyer’s Incentive to ReserveBuyer never reserves less than Buyer never reserves less than k*k*. In fact, she reserves . In fact, she reserves only if the reservation fee is below a certain threshold, only if the reservation fee is below a certain threshold, rrtt. . The reservation quantity will be at least The reservation quantity will be at least qqtt , , wherewhere

qqtt is a function ofis a function of ww and does not depend on and does not depend on pp

The buyer reserves iff The buyer reserves iff r r rrtt

)()( *0 kq Bt

B

)(

)()(

t

tt

qF

qFwpr Threshold fee

Threshold Reservation Qty.

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Supplier’s ProfitSupplier’s Profit

The The reservation reservation quantityquantity determines the final determines the final capacity, so we may rewrite the profit function ascapacity, so we may rewrite the profit function as

Supplier’s profit function is unimodal if the Supplier’s profit function is unimodal if the demand distribution has increasing failure rate demand distribution has increasing failure rate (IFR)(IFR)The function is strictly decreasing in The function is strictly decreasing in [k[koo,,))

),max(max wt qqq

q

sqq

dxxFqF

qFwpqVqScw

0

)()(

)()()()()(Max

max

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The Supplier’s Faces Three Profit Scenarios in The Supplier’s Faces Three Profit Scenarios in sequence as the sequence as the Buyer’s MarginBuyer’s Margin Decreases Decreases

)()(

)()(*

*

kSqS

kVqVcp

t

tt

)()( *kq It

I

rminr*

Reservation Fee

Exp

ec

ted

Pro

fit

Reservation FeeE

xpe

cte

d P

rofi

t

Reservation Fee

Exp

ec

ted

Pro

fit

rmin

rmin

Scenario 1: High Scenario 2: Medium

Scenario 3: Low

The buyer’s margin must be sufficiently high to justify capacity reservation

Theorem: It is individually rational for the supplier and the buyer to enter the reservation contract if the buyer revenue margin p is no less than the threshold pt as follows:

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Impact of Market Size and Variability: Impact of Market Size and Variability: Shifted Family DistributionsShifted Family Distributions

If the distribution is from a “shifted” familyIf the distribution is from a “shifted” family

the standard deviation remains the same as the market size increasesthe standard deviation remains the same as the market size increasesWith With linearlinear capacity cost capacity cost

the threshold reservation fee the threshold reservation fee rrtt does not change in market size does not change in market size the threshold reservation quantity the threshold reservation quantity qqtt does does increase in increase in (literature)(literature)

With With convexconvex capacity cost, capacity cost, bothboth the threshold fee the threshold fee rrtt and quantity and quantity qqtt increase in market size increase in market size there exists a threshold there exists a threshold tt over which reservation is not “win-win” over which reservation is not “win-win” the decrease in the coefficient of variation due to market size may not sufficient the decrease in the coefficient of variation due to market size may not sufficient

to offset the risk of expansion.to offset the risk of expansion.

),min(* wrr t

)0|()|( xFxF

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the standard deviation increases with the market size, thus, the standard deviation increases with the market size, thus, the threshold reservation fee the threshold reservation fee rrtt increasesincreases in market size in market size ..

even with linear capacity cost, the increase in reservation even with linear capacity cost, the increase in reservation payment (due to market size) is not sufficient to balance payment (due to market size) is not sufficient to balance the increase in riskthe increase in risk

Impact of Market Size and Variability: Impact of Market Size and Variability: Scaled Family DistributionsScaled Family Distributions

)1|()|(

xFxF

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Capacity reservation may create surplus for the Capacity reservation may create surplus for the channelchannel

However, except for a few special cases the However, except for a few special cases the surplus is sub-optimalsurplus is sub-optimal

We design two additional coordination We design two additional coordination mechanisms to achieve optimalitymechanisms to achieve optimality Partial-deduction (PD) contractPartial-deduction (PD) contract Cost-sharing (Options) contractCost-sharing (Options) contract

System Optimality and ReservationsSystem Optimality and Reservations

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Capacity Reservation with Partially Deductible FeesCapacity Reservation with Partially Deductible Fees

Suppose that the supplier offers a contract Suppose that the supplier offers a contract (r, r(r, r22)) where the reservation where the reservation fee is fee is r r and the deductible portion is and the deductible portion is rr22

TheoremTheorem. Under PD contract the buyer will reserve the channel optimal . Under PD contract the buyer will reserve the channel optimal capacity, i.e., capacity, i.e., q(r) = kq(r) = k00. Moreover,. Moreover,

(1) The buyer's profit is increasing in (1) The buyer's profit is increasing in rr,,(2) The supplier's profit is decreasing in (2) The supplier's profit is decreasing in rr,,(3) The supplier would only offer such contract if (3) The supplier would only offer such contract if vv0 0 2w 2wcc

thus,0but ,)( 22

rrwprv

cpr

o

cp

vwpr

vcp

vwp

o

o

o

)()(

marginal capacity cost at k0

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Cost-Sharing Contracts: Cost-Sharing Contracts: Capacity OptionsCapacity Options

Suppose the supplier proposes a contract where the buyer Suppose the supplier proposes a contract where the buyer pays pays V(Q) V(Q) for reserving for reserving Q Q units (options) whereas the units (options) whereas the supplier returns supplier returns rr22 for each exercised reservation quantity. for each exercised reservation quantity.

The channel is coordinated if The channel is coordinated if

Contract setting is independent of the demand distributionContract setting is independent of the demand distribution

*

0*

*

0

2 and )()(I

SI

I

Bwpcpr

*)1( getsSupplier I * getsBuyer I

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Capacity Reservation with Buyer CompetitionCapacity Reservation with Buyer Competition(Erkoc and Wu, 2002b)(Erkoc and Wu, 2002b)

Introducing horizontal buyer competitionIntroducing horizontal buyer competition

Still a supplier-lead single-period modelStill a supplier-lead single-period model

Two identical buyers from independent markets Two identical buyers from independent markets competing for the capacity competing for the capacity

The demand distribution for buyer The demand distribution for buyer ii is is GGii(x)(x)

The supplier offers a reservation fee, The supplier offers a reservation fee, rr, for each unit of , for each unit of capacity that is uniform across buyerscapacity that is uniform across buyers

Excess capacity, if any exists, is distributed Excess capacity, if any exists, is distributed proportionalproportional to the buyers’ reservation quantitiesto the buyers’ reservation quantities

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Integrated ChannelIntegrated Channel

Supplier

w p

c V(k)

Buyer i Market

Gi(x)

p

Buyer -i Market

G-i(x)

w

The buyers’ competition will be

•a Fixed Capacity Reservation (FCR) Game, if (total reservation amount) k*

•a Variable Capacity Reservation (VCR) Game, otherwise

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Fixed Capacity Reservation Game (FCR)Fixed Capacity Reservation Game (FCR)

Capacity is fixed at Capacity is fixed at k* k*

A zero-sum gameA zero-sum game

At equilibrium the buyers are compelled to reserveAt equilibrium the buyers are compelled to reserve

Capacity allocated to buyer Capacity allocated to buyer i:i:

With identical buyersWith identical buyers

*kqq

qh

ji

ii

)2/()(4

)( *2* kGkqGq

wpr

kk

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Capacity AllocationCapacity Allocation

q-i qi

Buyer -i demand, x-i Buyer i demand, xi

ii hx

iiii hkxhx *,

iiii hkxhx *,

iiii hxhx ,

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Variable Capacity Reservation (VCR) GameVariable Capacity Reservation (VCR) Game

Capacity is determined by total reservations: Capacity is determined by total reservations: k = qk = qi i + q+ q-i-i

Buyers’ profit function:Buyers’ profit function:

Optimal buyer profits decrease in reservation feeOptimal buyer profits decrease in reservation fee

At any point, each buyer wishes the other to At any point, each buyer wishes the other to reserve morereserve more

iq

iiqB dxxGrqqSwp0

)(),()(

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Game within a Game:FCR or VCR?Game within a Game:FCR or VCR?

rk rqrt

Bu

ye

r P

rofi

t

Reservation Fee

FCR

VCR

0.7

0.8

0.9

1

1.1

1.2

1 2 3 4 5 6 7 8 9 10

Reservation Fee

Res

erv

atio

n A

mo

un

t

Equilibrium Reservations, p=15,w=10,V(k)=5k,c=3,D~N(1,1/4)

FCR

VCR

In equilibrium, the buyers would prefer the VCR game so long as the reservation fee is below a certain threshold rt

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Supplier’s ProfitSupplier’s Profit

If the total reservation amount is If the total reservation amount is less less than than k*k*, the , the supplier’s optimal capacity will be supplier’s optimal capacity will be k* k* andand

Otherwise, the total reservation amount determines the Otherwise, the total reservation amount determines the final capacity andfinal capacity and

eqe

kFCRs dxxGqrkVkScw

0

** )()(2)()()(

eqeVCReeVCR

s dxxGqrqVqScw0

)()(2)2()2()(

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Supplier’s First Stage ModelSupplier’s First Stage Model

1,0

)(

0)1(

s.t.

)1( Max VCRs

FCRs

y

wqq

qyq

yy

t

qy=0 iff the surplus is created at the threshold reservation fee, i.e.,

)()( * tII qk

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TheThe N- N-Buyer CaseBuyer Case

Proportional allocation of excess capacityProportional allocation of excess capacity Becomes intractable Becomes intractable Existence of equilibrium can be established for a subset Existence of equilibrium can be established for a subset

of of uplifting contractsuplifting contracts

Uniform allocation of excess capacityUniform allocation of excess capacity

Equilibrium can be established via supermodularityEquilibrium can be established via supermodularity The equilibrium is uniqueThe equilibrium is unique

)),q(()q,( iiiiGiiVCRi xZqSqS

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ConclusionsConclusions

Impact of Impact of capacity reservationscapacity reservations on supply chain efficiency on supply chain efficiency (when are they beneficial?)(when are they beneficial?)

Convex capacity cost Convex capacity cost

Effects of market size and demand variabilityEffects of market size and demand variability

Horizontal buyer competitionHorizontal buyer competition

Proposed Contracts for channel coordinationProposed Contracts for channel coordination

Other ConsiderationsOther Considerations Voluntary ComplianceVoluntary Compliance Partial demand informationPartial demand information

Working papers Working papers (Erkoc and Wu, 2002 a,b)(Erkoc and Wu, 2002 a,b)

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Future ConsiderationsFuture Considerations

Multiple periods with inventory considerationsMultiple periods with inventory considerationsCorrelated demands and/or intra-market competitionCorrelated demands and/or intra-market competitionSequential or repeated negotiationsSequential or repeated negotiationsMultiple-supplier modelMultiple-supplier modelIn house production v.s. outsourcingIn house production v.s. outsourcingCapacity Reservation from the Capacity Reservation from the Contract ManufacturersContract Manufacturers PerspectivePerspective

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Buyer’s Profit and EquilibriumBuyer’s Profit and Equilibrium

i

i

i

i

h

hk

h

h

iik dydxxgygyKdxxKGxxgdxxxgqqS0

**

0)()()()()()(),(

*

* *

0

* )()()()()(K

h

xK

iiii

hGhGhdydxxgygyK

At equilibrium both buyers reserve the same quantity, qk yielding

0

0

** )()2/,2/()( B

q

keB

k

dxxGrKKSwp )2/(

4

)()( *2* KGK

r

wpqGq kk

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The Uplifting Contract (Spot Market Pricing)The Uplifting Contract (Spot Market Pricing)The supplier charges an extra The supplier charges an extra for any purchase order beyond the reservation for any purchase order beyond the reservation amount (spot market price)amount (spot market price)

The supplier offers a contract The supplier offers a contract ((,r,r))The contract coordinates the channel if The contract coordinates the channel if b,d b,d satisfy the following:satisfy the following:

With With dd the supplier dictates if the buyers choose to play the VCR game the supplier dictates if the buyers choose to play the VCR gameBy adjusting the reservation fee, the supplier may dictate which game (FCR or By adjusting the reservation fee, the supplier may dictate which game (FCR or VCR) the buyers playVCR) the buyers play

2/

0

)()(

)()()(2

1)2/(

)(ok

o

ooo

dxxkGxg

kFkGwpkrGrb

o/w

2/ if *

b

kqdb

)()2/(

)2/()(

*

*

rbkG

kGrrd